Fiber optical gyroscope

ABSTRACT

An optical gyroscope is provided for measuring a small angular difference. The gyroscope includes a laser, a pre-selection polarizer, a first beam splitter, a coil of optical fiber, a second beam splitter, a post-selection polarizer, a spectrometer and an analyzer. The laser emits a pulse beam of coherent photons. The beam has pulse duration σ. The pre-selection polarizer pre-selects the photons, and the first beam splitter separates the photons by their horizontal |+  and vertical |−  polarization eigenstates. These beams are launched into a fiber optical coil of radius r, which preserves polarization. The coil rotates by a difference rotation angle Δθ. The second beam splitter recombines the polarized photon beams as they exit the coil. The post-selection polarizer post-selects the photons. The spectrometer captures the post-selected photons and measures the associated frequency translation δω. The analyzer determines the difference rotation angle as 
                 Δ   ⁢           ⁢   θ     =       ±     (       c   ⁢           ⁢     σ   2     ⁢   tan   ⁢           ⁢   χ       2   ⁢   r       )       ⁢   δω       ,         
such that c is speed of light, and χ is post-selection polarization phase angle.

STATEMENT OF GOVERNMENT INTEREST

The invention described was made in the performance of official dutiesby one or more employees of the Department of the Navy, and thus, theinvention herein may be manufactured, used or licensed by or for theGovernment of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefor.

BACKGROUND

The invention relates generally to fiber optical gyroscopes. Inparticular, the invention relates to devices designed to resolve verysmall angular shifts by quantum weak measurements.

A fiber optical gyroscope (FOG) measures or maintains orientation,analogous to the function of a mechanical gyroscope based on theprinciples of conservation of angular momentum. The FOG's principle ofoperation is instead based on the interference of light that passesthrough a coil of an optical fiber, and contains no moving components.FOGs have been employed for inertial navigation in guided missiles.

In a conventional FOG, a laser emits a beam of photons that areseparated by a beam splitter into two polarized beams. Both beams enterinto a single optical fiber but in opposite directions. Due to theSagnac effect, the beam travelling against the rotation experiences aslightly shorter path delay than the other beam. Interferometry enablesthe resulting differential phase shift to be measured. This translatesone component of angular velocity into an interference pattern shiftthat can be measured photometrically.

Beam-splitting optics launches light from a laser diode into two wavespropagating in the clockwise and counter-clockwise directions through acoil consisting of many turns of optical fiber. The Sagnac effect'sstrength depends on the effective area of the closed optical path, andthus relates to the geometric area of the loop and the number of turnsin the coil.

SUMMARY

Conventional fiber optical gyroscopes yield disadvantages addressed byvarious exemplary embodiments of the present invention. In particular,resolution of small angles can be enhanced using quantum weakmeasurement by embodiments described herein to improve precision.

Various exemplary embodiments provide an optical gyroscope for detectingan angular difference. The gyroscope includes a laser, a pre-selectionpolarizer, a first polarizing beam splitter, a coil of optical fiber, asecond recombining beam splitter, a post-selection polarizer, aspectrometer and an analyzer. The laser emits a pulse beam of coherentphotons. The beam has pulse duration σ. The pre-selection polarizerpre-selects the photons, and the polarizing beam splitter separates thephotons by their horizontal |+

and vertical |−

polarization eigenstates.

The coil has radius r and preserves the polarization of photons. Thecoil rotates during said pulse duration σ by a difference rotation angleΔθ. After beam recombination by the second beam splitter, thepost-selection polarizer post-selects the photons. The spectrometercollects these photons and determines an associated frequency translatorδω. The analyzer determines the difference rotation angle as

${{\Delta\;\theta} = {{\pm \left( \frac{c\;\sigma^{2}\tan\mspace{11mu}\chi}{2r} \right)}{\delta\omega}}},$such that c is speed of light, and χ is a post-selection polarizationphase angle.

BRIEF DESCRIPTION OF THE DRAWINGS

These and various other features and aspects of various exemplaryembodiments will be readily understood with reference to the followingdetailed description taken in conjunction with the accompanyingdrawings, in which like or similar numbers are used throughout, and inwhich:

FIGURE is a schematic view of an AAV fiber optical gyroscope.

DETAILED DESCRIPTION

In the following detailed description of exemplary embodiments of theinvention, reference is made to the accompanying drawings that form apart hereof, and in which is shown by way of illustration specificexemplary embodiments in which the invention may be practiced. Theseembodiments are described in sufficient detail to enable those skilledin the art to practice the invention. Other embodiments may be utilized,and logical, mechanical, and other changes may be made without departingfrom the spirit or scope of the present invention. The followingdetailed description is, therefore, not to be taken in a limiting sense,and the scope of the present invention is defined only by the appendedclaims.

In accordance with a presently preferred embodiment of the presentinvention, the components, process steps, and/or data structures may beimplemented using various types of operating systems, computingplatforms, computer programs, and/or general purpose machines. Inaddition, those of ordinary skill in the art will readily recognize thatdevices of a less general purpose nature, such as hardwired devices, orthe like, may also be used without departing from the scope and spiritof the inventive concepts disclosed herewith. General purpose machinesinclude devices that execute instruction code. A hardwired device mayconstitute an application specific integrated circuit (ASIC) or afloating point gate array (FPGA) or other related component.

Various exemplary embodiments provide a fiber optical gyroscope thatuses the Aharonov-Albert-Vaidman (AAV) amplification effect. Photons ina laser pulse that have traversed a circular rotation sensing loop offiber and have their polarization states pre- and post-selected yield afrequency shift produced by the rotation of the fiber. A pure imaginaryweak value amplifies small frequency shifts from which small rotationangles are obtained. When ideal conditions prevail, rotation angles of˜750 femtoradians can be measured.

The weak value A_(w) of a quantum mechanical observable Â was describeda quarter century ago by Y. Aharonov et al., “How the Result of aMeasurement of a Component of the Spin of a Spin-½ Particle Can Turn Outto be 100”, Phys. Rev. Lett. 60, 14, 1351 (1988) and known as “AAV.” Seehttp://www.tau.ac.i1/˜vaidman/1vhp/m8.pdf for details.

This weak quantity is the statistical result of a standard measurementprocedure performed upon a pre- and post-selected (PPS) ensemble ofquantum systems when the interaction between the measurement apparatusand each system is sufficiently weak, i.e., when it is a weakmeasurement. Unlike a standard strong measurement of Â, whichsignificantly disturbs the measured system (i.e., it “collapses” thewavefunction), a weak measurement of observable Â for a PPS system doesnot appreciably disturb the quantum system and yields A_(w) as theobservable's measured value.

The peculiar nature of the virtually undisturbed quantum reality thatexists between the boundaries defined by the PPS states is revealed inthe eccentric characteristics of A_(w), namely that

(i) A_(w) can be complex valued;

(ii) Re A_(w) can lie far outside the eigenvalue spectrum limits ofoperator Â; and

(iii) the magnitude of Im A_(w) can be extremely large.

Characteristics (ii) and (iii) are referred to as the AAV amplificationeffect.

Experiments have verified several of the interesting unusual propertiespredicted by weak value theory. These include reports by N. Ritchie etal., “Realization of a Measurement of a ‘Weak Value’”, Phys. Rev. Lett.66, 1107 (1991); A. Parks et al., “An Optical Aharonov-Albert-VaidmanEffect”, Proc. R. Soc. A 454, 2997 (1998); K. Resch et al., “NonlinearOptics with Less than One Photon”, Phys. Leff. A 324, 125 (2004); Q.Wang et al., “Experimental demonstration of a method to realize weakmeasurement of the arrival time of a single photon”, Phys. Rev. A 73, 2,023814 (2006); K. Yokota et al., “Direct observation of Hardy's paradoxby joint weak measurement with an entangled photon pair”, New J. Phys.11, 033011 (2009); P. Dixon et al., “Ultrasensitive Beam DeflectionMeasurement via Interferometric Weak Value Amplification”, Phys. Rev.Leff. 102, 173601. Further, an AAV amplification of 10⁴ (i.e.,ten-thousand) has recently been achieved and used to observe thepreviously unobserved spin Hall effect of light, as reported by O.Hosten et al., in “Observation of the Spin Hall Effect of Light via WeakMeasurements”, Science 319, 787 (2008).

This disclosure describes a concept for a fiber optical gyroscope (androtation angle sensor)—the Aharonov-Albert-Vaidman effect fiber opticalgyroscope AAVFOG that uses the AAV amplification effect along withassociated findings recently supported by N. Brunner et al., “Measuringsmall longitudinal phase shifts: weak measurements or standardinterferometry?”, Phys. Rev. Lett. 105, 1, 010405 (2010) to amplifysmall rotations detected by a loop of fiber optical cable. Seehttp://arxiv.org/PS_cache/arxiv/pdf/0911/0911.5139v2.pdf for details.Under certain ideal conditions, rotation angles on the order of severalhundred femtoradians can be measured.

The following presents a brief review of weak measurement and weak valuetheory. Weak measurements arise in the von Neumann description of aquantum measurement at time t₀ of a time-independent observable Â thatdescribes a quantum system in an initial fixed pre-selected state |ψ_(i)

at t₀.

In this description, the Hamiltonian Ĥ for the interaction between themeasurement apparatus and the quantum system is:Ĥ=γ(t)Â{circumflex over (p)}.  (1)Here, interaction strength:γ(t)=γδ(t−t ₀),  (2)defines the strength of the measurement's impulsive interaction at t₀,{circumflex over (p)} is the momentum operator for the pointer of themeasurement apparatus, which is in the initial normalized state |φ

, and the Dirac delta function δ(t−t₀) models the interactioneffectively as an impulsive interaction between a photon of the beam andthe measurement apparatus at time t₀.

Let {circumflex over (q)} be the pointer's position operator that isconjugate to {circumflex over (p)}. Prior to the measurement, thepre-selected system and the pointer are in the tensor product state|ψ_(i)

|φ

. Immediately following the interaction, the combined system is in thestate:

$\begin{matrix}{{\left. \Phi \right\rangle = {{{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}{\int{\hat{H}{\mathbb{d}t}}}}\left. \psi_{i} \right\rangle\left. \varphi \right\rangle} = {{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}\gamma\hat{A}\hat{p}}\left. \psi_{i} \right\rangle\left. \varphi \right\rangle}}},} & (3)\end{matrix}$where use has been made of the fact that:∫ĥdt=γÂ{circumflex over (p)}.  (4)Also, one may note that i=√{square root over (−1)} is the imaginaryunit, and that

$\hslash = \frac{h}{2\pi}$represents the reduced Planck constant.

Here, the unitary evolution operator

${\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}\gamma\hat{A}\hat{p}}$in eqn. (3) is referred to as the von Neumann interaction operator. If astate |ψ_(f)

is then post-selected, the resulting pointer state for the PPS systemis:

$\begin{matrix}{{\left. \Psi \right\rangle \equiv \left\langle \psi_{f} \middle| \Phi \right\rangle} = {\left\langle {\psi_{f}{{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}\gamma\hat{A}\hat{p}}}\psi_{i}} \right\rangle{\left. \varphi \right\rangle.}}} & (5)\end{matrix}$Note that the probability of successfully obtaining a measured system inthe post-selected state |ψ_(f)

is |

ψ_(f)|ψ_(i)

|².

A weak measurement of Â occurs when the interaction strength γ issufficiently small so that the system is essentially undisturbed and theuncertainty Δq is much larger than Â's eigenvalue separation. In thiscase eqn. (5) becomes:

$\begin{matrix}{{{{\left. \Psi \right\rangle \approx {\left\langle \psi_{f} \right.\left( {\overset{\Cap}{1} - {\frac{\mathbb{i}}{\hslash}\gamma\hat{A}\hat{p}}} \right)\left. \psi_{i} \right\rangle\left. \varphi \right\rangle}} = {\left\langle \psi_{f} \middle| \psi_{i} \right\rangle\left( {1 - {\frac{\mathbb{i}}{\hslash}\gamma\; A_{w}\hat{p}}} \right)\left. \varphi \right\rangle}},}\;\quad} & (6)\end{matrix}$or else as:

$\begin{matrix}{\left. \Psi \right\rangle \approx {\left\langle \psi_{f} \middle| \psi_{i} \right\rangle{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}\gamma\; A_{w}\hat{p}}{\left. \varphi \right\rangle.}}} & (7)\end{matrix}$Here, the weak value A_(w) of the operator Â is defined by:

$\begin{matrix}{A_{w} \equiv {\frac{\left\langle \psi_{f} \right.\hat{A}\left. \psi_{i} \right\rangle}{\left\langle \psi_{f} \middle| \psi_{i} \right\rangle}.}} & (8)\end{matrix}$

The exponential operator in eqn. (7) is the translation operatorŜ(γA_(w)) for the initial normalized pointer state |φ

in the q representation. The translation operator is defined by theaction:

q|Ŝ (γA _(w))|φ

=φ(q−γA _(w)),  (9)which translates the pointer's wave-function over a distance γA_(w)parallel to the q-axis.

Weak measurements can be performed in position space or else in momentumspace. These measurements can be conducted upon an ensemble of identicalPPS systems. Based on the initial normalized pointer state |φ

, the initial mean pointer position is:q_(i)≡

φ|{circumflex over (q)}|φ

,  (10)and the initial mean pointer momentum is:p_(i)≡

φ|{circumflex over (p)}|φ

,  (11)whereas based on the final post-measured pointer state |Ψ

, the pointer's final mean post-measurement position is:q_(f)≡

Ψ|{circumflex over (q)}|Ψ

,  (12)and its final mean post-measurement momentum is:p_(f)≡

Ψ|{circumflex over (p)}|Ψ

.  (13)

The differences between these values provide information about weakvalue A_(w). R. Jozsa provides these differences in “Complex weak valuesin quantum measurement”, Phys. Rev. A 76, 044103 (2007), with details athttp://arxiv.org/PS_cache/arxiv/pdf/0706/0706.4207v1.pdf. For thegeneral case in which both the weak value A_(w) and the initial pointerstate φ(q) are complex valued, the translation in the mean pointerposition is:

$\begin{matrix}{{{{\delta\; q} \equiv {q_{f} - q_{i}}} = {{\gamma\mspace{11mu}{Re}\mspace{14mu} A_{w}} + {\left( \frac{m\;\gamma}{\hslash} \right)\left( \frac{{\mathbb{d}\Delta_{\varphi}^{2}}q}{\mathbb{d}t} \right){Im}\mspace{14mu} A_{w}}}},} & (14)\end{matrix}$and the translation in the mean pointer momentum is:

$\begin{matrix}{{{\delta\; p} \equiv {p_{f} - p_{i}}} = {2\left( \frac{\gamma}{\hslash} \right)\left( {\Delta_{\varphi}^{2}p} \right){Im}\mspace{14mu}{A_{w}.}}} & (15)\end{matrix}$Here, m is the mass of the pointer, Δ_(φ) ²q and Δ_(φ) ²p are theinitial pointer position and momentum variances, and the time derivativeis the rate of change of the initial pointer position variance justprior to t₀.

The FIGURE shows a schematic view 100 of an exemplary AAVFOG. The view100 shows the apparatus being disposed in the plane of the image androtating about point O. A laser 110 emits a coherent light beam 120,which passes through a pre-selection polarizer 130 and subsequentlythrough a first polarizing beam splitter 140.

The beam splitter 140 divides the pre-selected photons according totheir horizontal and vertical polarization states, |+

and |−

respectively. As shown, the horizontally and vertically polarized photonbeams traverse a polarization-preserving optical fiber in the form of acircular coil 150 having a radius r. The fiber 150 rotatescounter-clockwise indicated by direction arrows 160 through angle Δθaround point O, shifting the two beams along the circumference arc byΔr. The shifted beams are recombined by a second recombining beamsplitter 170 and pass through a post-selection polarizer 180 beforereaching a spectrometer 190 for detection of the frequency translatorδω.

In the AAVFOG in the schematic view 100, the laser 110 emits a beam 120as a pulse of temporal width σ, with the photon polarization beingpre-selected by the polarizer 130 before being intercepted by the firstbeam splitter 140, which segregates the pulse photons according to theirlinear polarization state. These photons are injected into the circularcoil 150 of radius r of the polarization preserving optical fiber suchthat photons in the horizontal polarization state |+

traverse the coil 150 unchanged in the clockwise (CW) direction, andphotons in the vertical polarization state |−

traverse the coil 150 unchanged in the counter-clockwise (CCW)direction.

The photons exit the fiber coil 150 and are recombined by the secondbeam splitter 170. The polarization states of the recombined collectionof photons are post-selected by the second polarizer 180 as they exitone of the output ports of the second beam splitter 170. These photonsare then analyzed by the spectrometer 190 to determine the frequencytranslator δωand the associated rotation angle Δθ.

For the case shown in the FIGURE in which the apparatus rotates in theCCW direction 160 about point O, during the time that the photons aretraversing the fiber, the coil 150 rotates through a difference angleΔθ=Δr/r. The photons in the horizontal state |+

travel an arc length Ar less than the half-circumference of the coil150, and arrive at the second beam splitter 170 at time earlier by:

$\begin{matrix}{{\tau = \frac{\Delta\; r}{c}},} & (16)\end{matrix}$than they would if no rotation occurred, as:Δθ=0.  (17)Similarly, the photons in vertical state |−

that travel an arc length Δr more than the half-circumference and arriveat the second beam splitter 170 at time later by

$\tau = \frac{\Delta\; r}{c}$from eqn. (16) later than they would if Δθ=0 from eqn. (17).

Thus, after the CW and CCW pulses are recombined by the second beamsplitter 170, the resultant emergent light is a superposition of the“early” pulse with photons in horizontal state |+

and the “late” photons with photons in vertical state |−

. This dynamic, which is introduced by the beam splitters 140 and 170and the fiber coil 150 of the apparatus is described for the CCWrotation by the von Neumann evolution operator

${\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}{\hat{A}}_{CCW}\hat{p}},$where:Â _(CCW)≡(−Δr)|+

+|+(Δr)|−

−|  (18)is the associated photon “which path” operator. Note that becauseÂ_(ccw) has length as its dimension, then interaction strength is unityas γ=1 in the CCW von Neumann operator.

To verify that Â_(CCW) produces the required state of light emergingfrom the second beam splitter 180, one can employ the facts that:(|±

±|)^(n)=|±

±|,  (19)and for n=1:|±

±|∓

∓|=0  (20)and observe that:Â _(CCW) ^(n)≡(−Δr)^(n)|+

+|+(Δr)^(n)|−

−|.  (21)

Applying eqn. (21) to the CCW von Neumann operator yields:

$\begin{matrix}{{{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}\gamma{\hat{A}}_{CCW}\hat{p}} = {{\sum\limits_{n = 0}^{\infty}{\frac{\left\lbrack {{- \frac{\mathbb{i}}{\hslash}}\hat{p}} \right\rbrack^{n}}{n!}{\hat{A}}_{CCW}^{n}}} = {{\sum\limits_{n = 0}^{\infty}{\frac{\left\lbrack {{- \frac{\mathbb{i}}{\hslash}}\left( {{- \Delta}\; r} \right)\hat{p}} \right\rbrack^{n}}{n!}\left.  + \right\rangle\left\langle +  \right.}} + {\sum\limits_{n = 0}^{\infty}{\frac{\left\lbrack {{- \frac{\mathbb{i}}{\hslash}}\left( {\Delta\; r} \right)\hat{p}} \right\rbrack^{n}}{n!}\left.  - \right\rangle\left\langle -  \right.}}}}},\mspace{79mu}{or}} & (22) \\{\mspace{79mu}{{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}\gamma{\hat{A}}_{CCW}\hat{p}} = {{{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}{({{- \Delta}\; r})}\hat{p}}\left.  + \right\rangle\left\langle +  \right.} + {{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}{({{- \Delta}\; r})}\hat{p}}\left.  - \right\rangle{\left\langle -  \right..}}}}} & (23)\end{matrix}$

Let the pulse serve as the measurement pointer with |φ

as its initial state and let:|ψ_(i)

=α|+

+β|−

  (24)be the pre-selected polarization state for the pulse photons. The actionof the von Neumann operator upon the tensor product state |ψ_(i)

|φ

produces the state:

$\begin{matrix}{{\left. \Psi \right\rangle = {{{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}\gamma{\hat{A}}_{CCW}\hat{p}}\left. \psi_{i} \right\rangle\left. \varphi \right\rangle} = {{{\alpha\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}{({{- \Delta}\; r})}\hat{p}}\left.  + \right\rangle\left. \varphi \right\rangle} + {{\beta\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}{({\Delta\; r})}\hat{p}}\left.  - \right\rangle{\left. \varphi \right\rangle.}}}}}\quad} & (25)\end{matrix}$

In the q representation, this becomes:

$\begin{matrix}{{\left\langle q \middle| \Phi \right\rangle = {{\alpha\left\langle q \right.{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}{({{- \Delta}\; r})}\hat{p}}\left. \varphi \right\rangle\left.  + \right\rangle} + {\beta\left\langle q \right.{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}{({\Delta\; r})}\hat{p}}\left. \varphi \right\rangle\left.  - \right\rangle}}},{or}} & (26) \\{{\Phi(q)} = {{{{\alpha\varphi}\left( {q + {\Delta\; r}} \right)}\left.  + \right\rangle} + {{{\beta\varphi}\left( {q - {\Delta\; r}} \right)}{\left.  - \right\rangle.}}}} & (27)\end{matrix}$This state in eqn. (27) is the required superposition of the “early”pulse with photons in horizontal state |+

, i.e., φ(q+Δr)|+

and the “late” pulse with photons in vertical state |−

, i.e., φ(q−Δr)|−

.

It is now advantageous to redefine the “which path” operator as:Â_(CCW)≡Δr{circumflex over (B)}_(CCW), (28)where:{circumflex over (B)} _(CCW)=−|+

+|+|−

−|,  (29)so that the CCW von Neumann operator can be rewritten as:

$\begin{matrix}{{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}{\hat{A}}_{CCW}\hat{p}} = {{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}\Delta\; r{\hat{B}}_{CCW}\hat{p}}.}} & (30)\end{matrix}$Note that now the arc shift Δr is effectively the interaction strength,such that:γ=Δr  (31)

If the arc shift Δr is sufficiently small, and also the pulse width:

$\begin{matrix}{{{\sigma »}\frac{\Delta\; r}{c}},} & (32)\end{matrix}$greatly exceeds the rotation time constant so that the measurement isweak, then substitution of eqn. (30) into eqn. (7) yields:

$\begin{matrix}{\left. \Psi \right\rangle \approx {\left\langle \psi_{f} \middle| \psi_{i} \right\rangle{\mathbb{e}}^{{- \frac{\mathbb{i}}{\hslash}}\Delta\;{r{(B_{CCW})}}_{w}\hat{p}}{\left. \varphi \right\rangle.}}} & (33)\end{matrix}$

After using eqn. (24) as the pre-selected polarization state and:|ψ_(f)

=η|+

+μ|−

,  (34)as the post-selected polarization state, one can determine from eqn. (8)that:

$\begin{matrix}{{\left( B_{CCW} \right)_{w} = {- \left( \frac{{\alpha\eta}^{*} - {\beta\mu}^{*}}{{\alpha\eta}^{*} + {\beta\mu}^{*}} \right)}},} & (35)\end{matrix}$where η* and μ* represent complex conjugates of the post-selectioncoefficients.

One can choose these PPS states such that:

$\begin{matrix}{{\alpha = {\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}}},{{{and}\mspace{14mu}\beta} = {\frac{\mathbb{i}}{\sqrt{2}} = \frac{{\mathbb{i}}\sqrt{2}}{2}}},} & (36)\end{matrix}$for the pre-selection, and

$\begin{matrix}{{\eta = {\frac{{\mathbb{i}\mathbb{e}}^{{\mathbb{i}}_{\chi}}}{\sqrt{2}} = \frac{{\mathbb{i}\mathbb{e}}^{{\mathbb{i}}_{\chi}}\sqrt{2}}{2}}},{{{and}\mspace{14mu}\mu} = {\frac{{\mathbb{e}}^{- {\mathbb{i}}_{\chi}}}{\sqrt{2}} = \frac{{\mathbb{e}}^{- {\mathbb{i}}_{\chi}}\sqrt{2}}{2}}},} & (37)\end{matrix}$for the post-selection, where χ is the polarization phase angle. Thisyields:(B _(CCW))_(w) =icot χ,  (38)as the pure imaginary weak value of {circumflex over (B)}_(CCW).

Because the pulse serves as the pointer, then application of eqn. (15)provides the following expression for the pointer translation inmomentum space:

$\begin{matrix}{{{\hslash\; k_{f}} - {\hslash\; k_{i}}} = {{\hslash\left( {k_{f} - k_{i}} \right)} = {{- 2}\left( \frac{\Delta\; r}{\hslash} \right)\left( {\hslash^{2}\Delta^{2}\; k_{i}} \right)\cot\mspace{11mu}{\chi.}}}} & (39)\end{matrix}$Here k_(i) and k_(f) are the initial and final wave numbers for thepulse, and Δ²k_(i) is the initial wave number variance. In order toconvert this into the associated frequency translation δω of the pulseas measured by the spectrometer, divide out the reduced Planck number ℏfactors, multiply both sides of the eqn. (39) by the speed of light c,and use the fact that:ω=ck  (40)to obtain:δω≡ω_(f)−ω_(f)−ω_(i)=−2c(Δr)(Δ² k _(i))cot χ.  (41)

Applying to eqn. (39) the fact that the spatial width cσ of the pulse isrelated to Δk_(i) according to:Δk _(i)=(cσ)⁻¹  (42)finally yields:

$\begin{matrix}{{{\delta\omega} = {{- 2}\left( \frac{\tau}{\sigma^{2}} \right)\cot\mspace{11mu}\chi}},} & (43)\end{matrix}$where τ is defined by eqn. (16).

The condition in which the apparatus rotates about point O in the CWdirection instead of the CCW direction is obtained by replacing Δr with−Δr, replacing −Δr with Δr (i.e., replacing ±Δr with ∓Δr), andsubstituting the subscript CCW by the counterpart CW for eqns. (18)through (35). This results in the following relationship between theweak values for the CCW and CW “which path” operators:(B _(CW))_(w)=−(B _(CCW))_(w).  (44)

Consequently, eqn. (43) can now be rewritten for both the CCW and CWrotation cases in the compact form:

$\begin{matrix}{{{\delta\omega} = {{\pm 2}\left( \frac{\tau}{\sigma^{2}} \right)\cot\mspace{11mu}\chi}},} & (45)\end{matrix}$where the “+” applies if the rotation is in the CW direction and the “−”applies if the rotation is in the CCW direction. In either case, theobserved frequency translation δω can be greatly amplified when χ issmall.

This enables determination of a very small τ or equivalently a verysmall rotation angle Δθ from the measured value of δω when pulse widthσ, polarization phase angle χ and coil radius r are fixed and known.Thus:

$\begin{matrix}{{\tau = {{\pm \left( \frac{\sigma^{2}\tan\mspace{11mu}\chi}{2} \right)}{\delta\omega}}},{or}} & (46) \\{{{\Delta\theta} = {{\pm \left( \frac{c\;\sigma^{2}\tan\mspace{11mu}\chi}{2r} \right)}{\delta\omega}}},} & (47)\end{matrix}$for determining values of time constant τ and measurable angledifference Δθ.

Values from Brunner are used to estimate Δθ. One can assume that theoperational wavelength of the pulsed laser is λ=700 nm (red light). Apulsed titanium (Ti): sapphire (Al₂O₃) laser operating at thiswavelength can generate a pulse with a 5×10⁻¹⁵ s temporal width andcurrently available spectrometers have a spectral resolution of about2×10¹⁰ Hz at λ=700 nm.

Using these values for σ=5×10⁻¹⁵ s and δω=2×10¹⁰ Hz , along with c=3×10⁸m·s⁻¹, χ=10⁻³ rad, and r=0.1 m, then eqn. (47) yields an ideal rotationangle of:Δθ_(ideal)=±7.5×10⁻¹³ rad,  (48)as an estimate for the achievable value of Δθ that can be measured under“ideal” conditions.

However, this result does not account for any required measurementintegration times. In order to obtain a crude estimate of theintegration time, assume that Δθ is stationary during the measurementintegration time interval T, and let I be the intensity associated withthe laser pulse.

Then the intensity reaching the spectrometer per pulse is:I

ψ_(f)|ψ_(i)

²=I sin² χ,  (49)where |

ψ_(f)|ψ_(i)

|² is the probability of post-selecting a photon in final state |ψ_(f)

. If I_(req) is the intensity required for a spectroscopic measurement,then the number N of pulses needed is approximately:

$\begin{matrix}{{N \approx \frac{I_{req}}{I\mspace{14mu}\sin^{2}\mspace{11mu}\chi}},} & (50)\end{matrix}$and if ρ is the pulse repetition rate, then the estimated measurementintegration time T_(est) is:

$\begin{matrix}{{T_{est} \approx \frac{N}{\rho}} = {\frac{I_{req}}{I\mspace{11mu}\rho\mspace{14mu}\sin^{2}\mspace{11mu}\chi}.}} & (51)\end{matrix}$

Assuming that:I_(req)≈I  (52)and using ρ≈10² MHz, and χ≈−3 rad in the eqn. (51) yieldsT_(est)≈10⁻² s.  (53)This suggests that precision Δθ measurements can be made by this deviceto provide direction and rate of change data at a frequency that rendersit viable for gyroscopic applications.

While certain features of the embodiments of the invention have beenillustrated as described herein, many modifications, substitutions,changes and equivalents will now occur to those skilled in the art. Itis, therefore, to be understood that the appended claims are intended tocover all such modifications and changes as fall within the true spiritof the embodiments.

What is claimed is:
 1. An optical gyroscope for detecting an angulardifference, said gyroscope comprising: a laser for emitting a pulse beamof coherent photons, said beam having pulse duration σ; a pre-selectionpolarizer for pre-selecting said coherent photons as polarized photons;a polarizing beam splitter for separating said polarized photons bytheir horizontal |+

and vertical |−

polarization eigenstates as separated photons; a coil of optical fiberhaving radius r, said coil being rotated by a difference rotation angleΔθ, said coil preserving polarization of said separated photons; arecombining beam splitter for recombining said separated photons fromsaid coil as recombined photons; a post-selection polarizer forpost-selecting said recombined photons as selected photons; aspectrometer for capturing said selected photons and measuringassociated frequency translation δω; and an analyzer for determiningsaid difference rotation angle as${{\Delta\theta} = {{\pm \left( \frac{c\;\sigma^{2}\tan\mspace{11mu}\chi}{2r} \right)}{\delta\omega}}},$wherein c is speed of light, and χ is polarization shift angle.
 2. Thegyroscope according to claim 1, wherein probability of post-selecting aphoton of said beam relates to said polarization phase angle χ as |

ψ_(f)|ψ_(i)

|²=sin² χ, such that initial state is |ψ_(i)

, and final state is |ψ_(f)

.